Implication Details

Claim: Given a morphism whose category has pushouts, if it is an extremal monomorphism, then it is a strong monomorphism.

Proof: Let $m : A \to B$ be an extremal monomorphism and consider a diagram $$\begin{CD} C @>{e}>> D \\ @V{f}VV @VV{g}V \\ A @>>{m}> B \end{CD}$$ in which $e : C \to D$ is an epimorphism. Choose a pushout $$\begin{CD} C @>{e}>> D \\ @V{f}VV @VV{u}V \\ A @>>{v}> P. \end{CD}$$ Here, $v$ is an epimorphism since $e$ is an epimorphism. Moreover, by the universal property of the pushout, there is a unique morphism $h : P \to B$ such that $h \circ v = m$ and $h \circ u = g$. Since $m$ is an extremal monomorphism, $v$ is an isomorphism. Then $v^{-1} \circ u : D \to A$ is the required filling of the first diagram, since $$v^{-1} \circ u \circ e = v^{-1} \circ v \circ f = f.$$

This implication has a dual.